Computational models of electromagnetic resonators: Analysis of edge element approximation

The purpose of this paper is to address some difficulties which arise in computing the eigenvalues of Maxwell's system by a finite element method. Depending on the method used, the spectrum may be polluted by spurious modes which are difficult to pick out among the approximations of the physically correct eigenvalues. Here we propose a criterion to establish whether or not a finite element scheme is well suited to approximate the eigensolutions and, in the positive case, we estimate the rate of convergence of the eigensolutions. This criterion involves some properties of the finite element space and of a suitable Fortin operator. The lowest-order edge elements, under some regularity assumptions, give an example of space satisfying the required conditions. The construction of such a Fortin operator in very general geometries and for any order edge elements is still an open problem. Moreover, we give some justification for the spectral pollution which occurs when nodal elements are used. Results of numerical experiments confirming the theory are also reported.